A new norm on the space of reproducing kernel Hilbert space operators and Berezin number inequalities
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912318168760320 |
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| author | Nayak, Raj Kumar Bhunia, Pintu |
| author_facet | Nayak, Raj Kumar Bhunia, Pintu |
| contents | In this note, we introduce a novel norm, termed the $t-$Berezin norm, on the algebra of all bounded linear operators defined on a reproducing kernel Hilbert space $\mathcal{H}$ as
$$\|A\|_{t-ber} = \sup_{ λ, μ\in Ω} \left\{ t|\langle A \hat{k}_λ, \hat{k}_μ\rangle| + (1-t) |\langle A^* \hat{k}_λ, \hat{k}_μ\rangle| \right\}, \quad t\in [0,1],$$
where $A \in \mathcal{B}(\mathcal{H})$ is a bounded linear operator. This norm characterizes those invertible operators which are also unitary.
Using this newly defined norm, we establish various upper bounds for the Berezin number, thereby refining the existing results. Additionally, we derive several sharp bounds for the Berezin number of an operator via the Orlicz function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_06754 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A new norm on the space of reproducing kernel Hilbert space operators and Berezin number inequalities Nayak, Raj Kumar Bhunia, Pintu Functional Analysis 47A30, 15A60, 47A12 In this note, we introduce a novel norm, termed the $t-$Berezin norm, on the algebra of all bounded linear operators defined on a reproducing kernel Hilbert space $\mathcal{H}$ as $$\|A\|_{t-ber} = \sup_{ λ, μ\in Ω} \left\{ t|\langle A \hat{k}_λ, \hat{k}_μ\rangle| + (1-t) |\langle A^* \hat{k}_λ, \hat{k}_μ\rangle| \right\}, \quad t\in [0,1],$$ where $A \in \mathcal{B}(\mathcal{H})$ is a bounded linear operator. This norm characterizes those invertible operators which are also unitary. Using this newly defined norm, we establish various upper bounds for the Berezin number, thereby refining the existing results. Additionally, we derive several sharp bounds for the Berezin number of an operator via the Orlicz function. |
| title | A new norm on the space of reproducing kernel Hilbert space operators and Berezin number inequalities |
| topic | Functional Analysis 47A30, 15A60, 47A12 |
| url | https://arxiv.org/abs/2504.06754 |